Its consistent with ZF that every cardinal strictly bigger than $\aleph_0$ is singular (Moti Gitik).
Is it provable in ZF that for every cardinal (initial von Neumann ordinal with respect to equinumerousity) number $\kappa$ there is a cardinal $\lambda$ such that every $\in$-confinal subset of $\lambda$ is strictly bigger in cardinality than $\kappa$?
The cofinality of an ordinal $\lambda$ is the least ordinal $\kappa$ such that there is an unbounded subset of order type $\kappa$ in $\lambda$.
An immediate consequence of the definition is that if $\kappa$ is the cofinality of any ordinal $\lambda$, then the cofinality of $\kappa$ is $\kappa$.
And an immediate consequence of that is that cofinality is always a cardinal. So we can replace "order type $\kappa$" by "cardinality $\kappa$".
And an immediate corollary to all of that is that if every uncountable cardinal is singular, then the only possible cofinality is the only cardinal that $\sf ZF$ proves to be regular: $\aleph_0$. Therefore, in this case, every limit ordinal must have a countable cofinal sequence.